1&2 Propositional Calculus Propositional Calculus Problems Jim Woodcock University of York October 2008 1. Let p be it s cold and let q be it s raining. Give a simple verbal sentence which describes each of the following propositions: (a) p (b) p q (c) p q (d) q p (e) p q (f) q Propositional Calculus 3&4 2. Let p be she s tall and let q be she s beautiful. Write propositions that symbolise the following: (a) She s tall and beautiful. (b) She s tall but not beautiful. (c) It s false that she s short or beautiful. (d) She s neither tall nor beautiful. (e) It isn t true that she s short or not beautiful. 3. Translate into symbols the following compound statements: (a) We ll win the election, provided that Brown is elected leader of the party. (b) If Brown isn t elected leader of the party, then either Straw or Johnson will leave the cabinet, and we ll lose the election. (c) If x is a rational number and y is an integer, then z isn t real. (d) Either the murderer has left the country or somebody is hiding him. (e) If the murderer has not left the country, then somebody is hiding him. (f) The sum of two numbers is even if and only if either both numbers are even or both numbers are odd. (g) If y is an integer, then z isn t real, provided that x is a rational number.
5&6 Propositional Calculus 4. Let p be the proposition It s snowing. Let q be the proposition I ll go to town. Let r be the proposition I have time. (a) Using propositional connectives, write a proposition which symbolises each of the following: i. If it isn t snowing and I have time, then I ll go to town. ii. I ll go to town only if I have time. iii. It isn t snowing. iv. It s snowing, and I ll not go to town. (b) Write a sentence in English corresponding to each of the following propositions: i. q r p ii. r q iii. (q r) (r q) iv. (r q) Propositional Calculus 7&8 5. State the converse and contrapositive of each of the following: (a) If it rains, I m not going. (b) I ll stay only if you go. (c) If you get 4lbs, you can bake the cake. (d) I can t complete the task if I don t get more help. 6. Let p, q, r denote the following statements: p: Triangle ABC is isosceles; q: Triangle ABC is equilateral; r: Triangle ABC is equiangular. Translate each of the following into an English sentence. (a) q p (b) p q (c) q r (d) p q (e) r p
9&10 Propositional Calculus 7. (a) How many rows are needed for the truth table of the following proposition? p q r s t (b) Ifp1, p2,...,pn are propositional variables, and the compound statement p contains at least one occurrence of each propositional variable pi, how many rows are needed in order to construct the truth table for p? 8. Find truth tables for the following propositions: (a) p q (b) (p q) (c) (p q) 9. Express each of the following statements in the form p q. (a) Rain on Tuesday is a necessary condition for rain on Sunday. (b) If it rains on Tuesday, then it rains on Wednesday. (c) But it rains on Wednesday only if it rains on Friday. (d) Moreover, no rain on Monday means no rain on Friday. (e) Finally, rain on Monday is a sufficient condition for rain on Saturday. Given that it rains on Sunday, what can be said about Saturday s weather? 10. Verify that the proposition p (p q) is a tautology. 11. Verify that the proposition (p q) (p q) is a contradiction. Propositional Calculus 11&12 12. Which of the following propositions are tautologies? (a) p (q p) (b) q r ( r q) (c) (p q) ((q r) (r p)) (d) (p (q r)) ((p q) r) 13. Show that the following pairs of propositions are logically equivalent. (a) p q, q p (b) (p q) r, (p r) (q r) (c) p q r, r q p (d) p q r, (p q) r 14. Show that the proposition ( p q) (p q) isn t a tautology. 15. Show that the following argument is valid: p q, q p. 16. Show that the following argument is valid: p q, q p. 17. Show that the following argument is a fallacy: p q p q. 18. Test the validity of the following argument: If I study, then I ll not fail mathematics. If I do not play Lemmings, then I ll study. But I failed mathematics. Therefore I played Lemmings. 19. What conclusion can be drawn from the truth of p p?
13&14 Propositional Calculus 20. For each of the following expressions, use identities to find equivalent expressions which use only and and are as simple as possible. (a) p q r (b) p ( q r p) (c) p (q r) 21. For each of the following expressions, use identities to find equivalent expressions which use only and and are as simple as possible. (a) p q r (b) (p q r) p q (c) p q ( r p) Propositional Calculus 15&16 22. Establish the following tautologies by simplifying the left-hand side to the form of the right-hand side: (a) (p q p) true (b) ( (p q) p) false (c) ((q p) ( p q) (q q)) p (d) ((p p) ( p p)) false 23. (a) The nand operator (also known to logicians as the Sheffer stroke), is defined by the following truth table: p q p nand q t t f t f t f t t f f t Of course, nand is a contraction of not-and; p nand q is logically equivalent to (p q). Show that i. (p nand p) p ii. ((p nand p) nand (q nand q)) p q iii. ((p nand q) nand (p nand q)) p q
17&18 Propositional Calculus (b) Find equivalent expressions for the following, using no connectives other than nand: i. p q ii. p q (c) The nor operator (also known to logicians as the Pierce arrow), is defined by the following truth table: p q p nor q t t f t f f f t f f f t For each of the following, find equivalent expressions which use only the nor operator. i. p ii. p q iii. p q Propositional Calculus 19&20 24. Formalise the second and third proofs of the theorem: if x 2 3x + 2 < 0, then x > 0 25. Write a program to construct truth tables of propositions.